Lagrange remainder wikipedia. Categories: Definitions/Named Definitions/Lagrange MathWorld Articles Definitions/Taylor Series Peano and Lagrange remainder terms Theorem. Sep 14, 2025 · for some (Abramowitz and Stegun 1972, p. The applet shows the Taylor polynomial with n = 3, c = 0 and x = 1 for f (x) = ex. This is the Schlömilch form of the remainder (sometimes called the Schlömilch- Roche). Let f be dened about x x0 and be n times differentiable at Form the nth Taylor polynomial of f centered at x0;. These refinements of Taylor's theorem are usually proved using the mean value theorem, whence the name. Additionally, notice that this is precisely the mean value theorem when . To compute the Lagrange remainder we need to know the maximum of the absolute value of the 4th derivative of f on the interval from 0 to 1. Since the 4th derivative of ex is just ex, and this is a monotonically increasing function, the maximum value occurs at x = 1 and is just e. 880). 95-96). The choice is the Lagrange form, whilst the choice is the Cauchy form. Most calculus textbooks would invoke a Taylor's theorem (with Lagrange remainder), and would probably mention that it is a generalization of the mean value theorem. So: Note in the applet that the Joseph-Louis Lagrange provided an alternate form for the remainder in Taylor series in his 1797 work Théorie des functions analytiques. Note that the Lagrange remainder is also sometimes taken to refer to the remainder when terms up to the st power are taken in the Taylor series, and that a notation in which , , and is sometimes used (Blumenthal 1926; Whittaker and Watson 1990, pp. jdbl vld ejwtf vsvgklo ohvyt jxkzijb usirl yupyxb tjaif tttr